Approximation of stochastic differential equations driven by α-stable Lévy motion

Aleksander Janicki; Zbigniew Michna; Aleksander Weron

Applicationes Mathematicae (1997)

  • Volume: 24, Issue: 2, page 149-168
  • ISSN: 1233-7234

Abstract

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In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.

How to cite

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Janicki, Aleksander, Michna, Zbigniew, and Weron, Aleksander. "Approximation of stochastic differential equations driven by α-stable Lévy motion." Applicationes Mathematicae 24.2 (1997): 149-168. <http://eudml.org/doc/219159>.

@article{Janicki1997,
abstract = {In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.},
author = {Janicki, Aleksander, Michna, Zbigniew, Weron, Aleksander},
journal = {Applicationes Mathematicae},
keywords = {α-stable Lévy motion; convergence of approximate schemes; stochastic differential equations with jumps; stochastic modeling; heavy-tailed claims; -stable Lévy motion; collective risk theory; insurance company; weak approximations; ruin probability},
language = {eng},
number = {2},
pages = {149-168},
title = {Approximation of stochastic differential equations driven by α-stable Lévy motion},
url = {http://eudml.org/doc/219159},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Janicki, Aleksander
AU - Michna, Zbigniew
AU - Weron, Aleksander
TI - Approximation of stochastic differential equations driven by α-stable Lévy motion
JO - Applicationes Mathematicae
PY - 1997
VL - 24
IS - 2
SP - 149
EP - 168
AB - In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to α-stable Lévy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time α-stable model of cumulative gain in the Duffie-Harrison option pricing framework.
LA - eng
KW - α-stable Lévy motion; convergence of approximate schemes; stochastic differential equations with jumps; stochastic modeling; heavy-tailed claims; -stable Lévy motion; collective risk theory; insurance company; weak approximations; ruin probability
UR - http://eudml.org/doc/219159
ER -

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